Energy functionals -
The previous paragraphs summarized important general concepts of the EDF
theory. In practical applications, one must specify the effective Hamiltonian
\(\hat{H}_{\rm eff}\)  or, alternatively, the energy functional
\(E[\rho ,\kappa ,\kappa ^*]\)  in order to perform calculations. The Skyrme and
Gogny energy functionals are among the most commonly used models for nuclear
forces [1]}, [2]}, [3]}, [4]}. These functionals
are derived from non-relativistic, two-body effective Hamiltonians. The Skyrme
Hamiltonian is a local zero-range potential, i.e., of the type
\(\hat{V}(\mathbf {r}_1,\mathbf {r}_2)\delta (\mathbf {r}_1-\mathbf {r}_2)\)  with \(\mathbf {r}_1\) 
and \(\mathbf {r}_2\)  the coordinates of the two nucleons; the Gogny Hamiltonian is a
local, finite-range potential of the type \(\hat{V}(\mathbf {r}_1,\mathbf {r}_2)\) 
where the dependency on \(\mathbf {r}_1\)  and \(\mathbf {r}_2\)  is proportional to
\(\exp \big [(\mathbf {r}_1 - \mathbf {r}_2)^2/\mu ^2 \big ]\)  with \(\mu \)  a typical length
scale. Each of these Hamiltonian contains a central part, a density-dependent
part that mocks up the effect of three-body forces and a spin-orbit part. In
addition, both effective Hamiltonians contain a Coulomb potential. While the
form (REF ) suggests that the same potential should be used to
compute both the mean field energy (proportional to \(\rho \rho \) ) and the pairing
energy (proportional to \(\kappa \kappa ^*\) ), practitioners tend to adopt the
following, more flexible (and less consistent) formula
\(E[\rho ,\kappa ,\kappa ^{*}] =\sum _{ij} t_{ij} \rho _{ji}+\frac{1}{2} \sum _{ijkl} \bar{v}_{ijkl}^{\rho \rho }\rho _{lj}\rho _{ki}+\frac{1}{4}\sum _{ijkl} \bar{v}_{ijkl}^{\kappa \kappa } \kappa ^{*}_{ij}\kappa _{kl}.\) 
